Properties

Label 97104bi
Number of curves $4$
Conductor $97104$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bi1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 97104bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97104.bc4 97104bi1 \([0, -1, 0, 4528, -31686720]\) \(103823/4386816\) \(-433713454654685184\) \([2]\) \(1327104\) \(2.0632\) \(\Gamma_0(N)\)-optimal
97104.bc3 97104bi2 \([0, -1, 0, -1475152, -676827200]\) \(3590714269297/73410624\) \(7257923592736997376\) \([2, 2]\) \(2654208\) \(2.4097\)  
97104.bc2 97104bi3 \([0, -1, 0, -3139792, 1132969408]\) \(34623662831857/14438442312\) \(1427492444399286386688\) \([2]\) \(5308416\) \(2.7563\)  
97104.bc1 97104bi4 \([0, -1, 0, -23485392, -43799289408]\) \(14489843500598257/6246072\) \(617533414928252928\) \([2]\) \(5308416\) \(2.7563\)  

Rank

sage: E.rank()
 

The elliptic curves in class 97104bi have rank \(2\).

Complex multiplication

The elliptic curves in class 97104bi do not have complex multiplication.

Modular form 97104.2.a.bi

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2q^{5} - q^{7} + q^{9} - 6q^{13} - 2q^{15} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.